Remath teacher scenarios

1. Dissemination of the ReMath Scenarios. Scenarios for teachers,
scenarios by teachers.
This work has been led by C. Kynigos with contribution of K. Makri in the theoretical
argumentations and G. Psycharis, S. Keisoglou and K. Gavrilis in the design and
implementation of the scenario template.
Introduction
In ReMath so far, we have designed and developed scenarios with a dual aim to a)
exemplify the educational goals of our research experiments and b) to provide data
for the building of the ITF through the elaboration of the ways in which theoretical
frameworks have been operationalised, contexts of design and use have been made
explicit and the intended mediation of representations has been made clear.
Through the conceptual model developed by the ITD group, the PPM tool has been
built providing a hierarchical structure to scenario descriptions, dynamic
representation and multiple views of the structure.
It has been suggested by our reviewers to disseminate the results of ReMath to
wider communities than the research one. In that wake it was explicitly suggested
we develop descriptions of the scenarios understood and widely useable by
teachers. What follows is a suggested rationale for developing such scenarios with
respect to their functionality within teacher communities. These new scenarios are
portrayed as pragmatic and practical, yet very different to classical activity plans or
systemic curricula. They are presented through a rational for their use to empower
teachers by adopting a perspective of including them in the design of innovations,
rather than providing them with the role of implementers of exemplary curricula.
We suggest a scenario template derived from the PPM scenarios, but containing a
simplification and more pragmatic nature of theoretical frameworks and rationale,
pointing to the relevant aspects of context and the ways in which the
representations are to be used. There is more explicit mapping onto specific
curricula but also the ground for linking the scenario onto a broader conceptual area.

2. The template differs from a classical activity plan because it has contextual and
argumentation data. The argumentation data are related to pedagogy. The
contextual data refers to what needs to be explicit about the learning environment.
It is suggested that we use the project site to disseminate these scenarios. However,
it would also be great if the PPM tool could be extended to incorporate such
scenarios, especially with wiki-style co-editing features and some basic annexing.
Properly organizing a large quantity of teacher-developed scenarios would require
sources we need to seek outside ReMath.
What are these new scenarios.
From the moment the ReMath scenarios are addressed to teachers, they will inevitably take
on new meanings related to perspectives of curriculum and lesson plans resident in teachers’
professional lives. Curriculum theory and practice has been a constituent element of formal
education since its foundations. Directly related to other schooling ideas such as subject and
lesson, our current appreciation of curriculum emerged in the school (Smith, 1996, 2000). Its
status and stability as an institutional form have inspired metaphors of curriculum as inherent
to the “deep grammar” of schooling or as part of the “DNA of classroom life” (Cuban, 1986,
Tyack & Tobin, 1994, in Ruthven, 2007). An established assumption about the function of
formal curricula is based on Ralph Tyler’s “curriculum as product” concept. Tyler, on the basis
of previous work by Franklin Bobbit (1918, 1928) emphasised on rationality and relative
simplicity as the basic merits in describing and managing instruction. Education, under this
lens, is viewed as a technical exercise, heavily influenced by the development of management
thinking and practice: objectives are set, a plan drawn up, then applied, and the outcomes (as
products) are measured (Smith, 1996, 2000). The approach of curriculum as fixed and the
teacher as its conduit, rather than as a user or designer is rooted in the historical assumption
that knowledge can be transmitted through teachers from those who know to those who do
not (Olson, 2000, her emphasis). Its logic is for curriculum to be designed outside the
classroom or school, as is the case with the National Curriculum in the UK, turning educators
into technicians, who apply programs and are judged by their actions (Smith, 1996, 2000, our
emphasis). Similar initiatives are witnessed in the US: recent calls for centralized curricula,
system-wide testing regimes, the standards movement, etc. (Hargreaves, 2000).

3. Digital and communicational media, however, invite both pedagogical knowledge and thinking
‘out of the box’ with respect to new kinds of learning activity, new means for expression, new
kinds of social structures, new temporal and spatial conditions for learning (Kynigos, 2004).
The management of technology-supported classroom environments is logistically difficult.
Students have more autonomy and work both individually and in a variety of collectives. There
is of course a large spectrum of social orchestrations and teacher epistemologies and teaching
paradigms ranging from more traditional curriculum deliveries to student-driven classroom
work where teachers need to provide individual guidance to several students simultaneously,
and coordinate students who work at different paces (Edelson, 1998). In all cases however, the
use of digital tools in the mathematics class creates a perturbation in teachers’ habits of work
(Laborde ++).
Ruthven (2007) summarises these kinds of perturbations in the following points:
• Working environment: disruption to normal working practices posing additional
organisational demands on the teacher
• Resource system: limited scope for teacher adaptation and reorganisation of CAI materials
• Activity format: disruption of familiar activity formats and their supporting classroom
routines-prototypical structures
• Curriculum script: official curricula recognising new tools without anticipating implications
for changed curriculum sequence
• Time economy: time and cost associated with innovation itself
So, curriculum development for innovations with the use of digital technologies is a complex
enterprise. Goldenberg (1999:191) describes curriculum development integrating the use of
technology as “a long, complex business that involves many different kinds of contributions […]
and a wide variety of activities that interact in complicated ways”. Innovative curriculum will
not work if it is delivered to teachers to implement. It cannot be done for teachers but needs
to include them in the design and development process, as Noss and Hoyles (1996, p. 186), put
it, to “design innovation with, rather than for, the teachers”. In ReMath, the objective was to
disseminate the innovations designed through the academic scenarios to teacher
communities. The task was thus to transpose the scenarios from academic documents into

4. malleable pieces of curricula so that teachers would feel invited to reflect over, transpose
them themselves and design new ones using these as starting points or idea generators.
Scenarios can thus be thought of as new formats for learning with new technologies,
the production of which cannot be done without the input from teachers. They are
designed to have the following kinds of functionality:
• To make implicit aspects of the learning environment explicit
• To get designers to think “out of the box”
• To provide an arena for reflection on new kinds of learning
• To inspire creative ideas on using digital media for learning
• To identify new ways of learning traditional concepts and/or new concepts
made available for generating rich learning environments.
These aspects of scenarios were derived from work within the ‘FOMAT’ ERT at
Kaleidoscope and seem to be interestingly cohesive with the PPM and the scenario
conceptual model developed by the ITD group. The task is how to change the
existing scenarios presently addressing the research community, so that they can
become operational for teachers without losing these characteristics.
Who will be developing them and why.
Teachers’ decision making is portrayed in the literature as a basic skill, decisive in effecting
positive learning outcomes (Shavelson, 1973, Hasselbrig & Gray, 1984). It is also considered as
a characteristic of flexible and creative teachers (Cooney, 1988). Teachers make decisions in a
time continuum including all phases of teaching (pre-active, interactive and post-active
(Shavelson, 1973), addressing cognitive, affective and managerial issues (Cooney, 1988).
These decisions are made in a process of curriculum development, in three levels: design,
construction, and curriculum mapping (Remillard, 1999). In practice, many teachers actively
reflect and reformulate their practice, try out innovations, and are by definition sensitive to
the context of the classroom and their students (Kynigos, 2003, 2007). When addressed with
the challenge of integrating digital media in their lessons, they plan their own educational

5. activities and resourcefully invent problem situations, project themes, artefacts and tools to
support student learning (ibid). This activity, however, remains tacit and unrecognised.
One reason for this is the conservative nature of the organisational context of the profession.
At the systemic level, the role apparently prescribed for teachers is that of content delivery,
prescribing a busy ant-like existence for the teacher’ s day to day job (ibid). Any attempt to
integrate digital media in teaching implicitly questions assumptions on the function and status
of established schooling norms. Formal curricula, for instance, though recognising the use of
new tools, don’t anticipate practical implications (Ruthven, 2007). For teachers to design
creative activities with technology, they would have to disturb fixed curricular sequences.
Teachers are both unprepared and not supported by organisational or systemic mechanisms
both to grapple with the complexities of design and to reflect on this process.
Dissemination of the ReMath scenarios to teachers and teacher communities has
more chances of succeeding if, rather than merely making them available as pre-
prescribed pieces of curricula in the traditional sense, to make them available as
objects for reflection, artefacts for instrumentation and instrumentalization (I’m using
Rabardel’s words here for scenarios instead of DDAs), ideas for design, and boundary
objects for communication.
This is line with what Jaworski (2003, 2004) and Wells (1999) have termed as the “inquiry
approach to curriculum design”. According to this, education has a dialogic character:
procedures, programs and curricula are constantly revised, as objects for negotiation. The
teacher is therefore considered as a reflective practitioner who shapes professional practice
through critical reflection (Schon 1987, Eisner 1998) and as an active creator of personal
pedagogy, directly involved in planning and implementing innovation (Hoyles, 1992, Κynigos,
2003, Kynigos et Argyris, 2004, Kynigos, in press). Design is viewed as a supporting mechanism
for reflection, and as a means for expressing and making explicit creative ideas on pedagogy.
How might they be used.
In the wake of the above arguments, teacher scenarios in ReMath can be used as
malleable personalized curricula. Each one of the 13 scenarios can be written in
collaboration with our teacher – researchers and mediated as an example of how the
corresponding DDA can be used for a course in a specific mathematical topic (or
more broadly a conceptual field) corresponding to a particular class in a particular

6. country. In that sense, the scenario will be read by outsider teachers as a practical,
specific example which however leaves room for them to transpose into their
curriculum and classroom pragmatics. It should thus be detailed and precise enough
so as to make concrete sense but on the other hand not too grounded on a specific
curriculum so that teachers from others can find it useful.
Furthermore, these scenarios can be used as starting points for teachers to engage in
scenario design and development within teacher communities of practice and within
teacher education initiatives. In that sense they are expected to be shareable,
questionable and malleable rather than presented as some kind of new trans-
European fixed curriculum.
What will be their functionality from a socio-systemic point of view.
In order to consider the ReMath scenarios as artifacts designed to support the
generation of teacher communities of practice we need to look at the wider picture
of how they might be operationalised in such communities across Europe. Adopting
the approach of teachers as co-designers of innovations with the use of digital media
would mean that our concern would be for these 13 scenarios to have a good chance
of being used to generate a large number of others by teachers who share,
communicate and reflect over them.
Communal design can generate the need to be explicit, to reflect and to express
meanings through argumentation. During a collaborative activity, a community
works towards a common goal, which can be an ideal object to be created, or a
specific improvable object (Bereiter &Scardamalia, 2003). This object is both the
centre of the activity, and also functions as a communicational tool to shape a
common language within the community. The notion of ‘boundary objects’
(originally coined by Star and Griesemer, 1989) has recently been discussed in
reference to digital artifacts and the ways they are used to generate communication
over mathematical ideas which are otherwise bound to the situations in which they
emerged between diverse communities such as learners and teachers (Hoyles et al,
2004). Cobb et. al. (2003) extend this notion by proposing that the term “boundary
objects” can refer to specific objects within different communities, which are

7. «relatively transparent means of conveying meaning among the members of the
community who created them». They can also be the centre around which
community members organize their activities and can additionally operate as tools
for communication among the members of the same community, and the members
of other communities. The ReMath scenarios can be designed as improvable
boundary objects and the process of changing them can be orchestrated to enhance
integration in scenario design.
How do they fit and/or serve the building of the ITF language.
In the framework of addressing the problem of fragmentation of theoretical
frameworks on learning mathematics with ICT, the building of the ITF language
aimed at perceiving theory in terms of functionality (Del, 9), i.e. specific ways in
which it can be implemented in educational practices. This is why the notion of
‘concerns’ became a primary notion for elaborating the ways in which theoretical
frameworks drove the design and uses of DDAs during research experiments. These
concerns are not parameters taken into account in a positivistic way. Rather, they
are issues related to context and the mediation of representations which usually
remain tacit in research, but which have an important effect in the negotiation of
meanings, the roles and the kinds of uses of the DDAs in the classroom.
In that sense, the research version of our ReMath scenarios is an attempt to
elaborate the didactical functionality of theoretical frameworks at the level of
design. This elaboration however, is mediated by researchers to researchers. The
framework above discusses ways in which an attempt to transpose these
descriptions into descriptions made by teacher-researchers to teachers may have
chances of generating an operationalisation of these scenarios within teacher
communities. The aim is to go beyond a scenario description, albeit improved with
respect to its understandability and familiarity for teachers, but still written by
researchers with the aim for teachers to perceive it as an exemplary form of
curriculum to be implemented. Instead the aim is for the new scenario descriptions
to invite teachers to engage with scenario design for themselves, to reflect on
theoretical and practical issues in using the technology and to take part in

8. communities of practice either in a professional or in a professional development
capacity. In that sense, the transposition of these scenario descriptions becomes a
greater challenge in the process of building the ITF language, i.e. it requires a
mediation of that language so that it becomes operational to teachers. The point is
to make it attractive to teachers without losing the rigor, depth and range of issues
needing to be addressed so that implementation of a scenario has good chances of
providing so educational added value.
The scenario template.
One of the ways in which this aim can be addressed is if the ways in which the
scenarios are described have some degree of uniformity in structure, topics
addressed and the kind of language used. The ETL and CTI groups came up with this
objective within their roles in a National Initiative for teacher education in the uses
of technology in the teaching of four disciplines, amongst which was mathematics.
The initiative consists firstly of the training of experienced and technology literate
teachers (a 350 hour course) to become teacher educators, effected in a number of
University sites (ETL amongst them). Secondly it consists of those teacher educators
providing courses (96 hours) to a total of 15000 teachers (8000 in the first phase).
Through CTI, an academic committee has been designing the courses, preparing
content and methods of evaluation. It is though this capacity that the challenge of
creating a framework for the description of scenarios arose. Before this venture we
had developed a framework for scenario description but felt it was lacking in
relevance to teachers, it was far too academic. We thus used the general work done
at the FORMAT ERT within Kaleidoscope and the work done in TELMA and ReMath to
try to develop a template for describing scenarios which could become an artifact of
the kind described in the above framework. We tried to simplify the ‘theoretical
framework’ and ‘rationale’ sections of the ReMath academic scenario versions by
introducing sections with specific practical topics (see section 3). We also tried to
provide equivalent section titles which would give a sense of context (section 4). In
section 1 we aimed at getting teachers to feel ownership of scenarios which they

9. might design and to immediately map them onto their own curriculum as part of the
scenario identity. However, we also gave a section titled ‘subject area’ to give the
chance of describing the wider area of mathematical topics, so that teachers from
different contexts would not necessarily alienate themselves from lack of capability
to map it onto their own curriculum. Section 5 which is meant to be the main length
of the scenario addresses the different aspects pertinent to the enactment of the
activity. It is important to say that we emphasize description of how the tools are to
be used, the teachers’ didactical rationale and the social orchestration. We also have
a section describing implementations of this scenario if there are any and hints from
teacher to teacher of what might be crucial for its success. The section describing
extensions aims at teacher to question the scenario and consider it a as a case for
something more general. In the reference section, other scenarios could be
mentioned providing opportunities for grouping scenarios together. Finally another
thing we thought would make scenario design more popular amongst teachers was
to make the structure flat. We acknowledge that this impoverishes the
representational capacity of a scenario but on the other hand, to get teachers
engaged with using, developing and sharing scenarios we felt that a format familiar
to them was more appropriate at this phase. Once we get teacher communities
going, then a more complex structure could be gradually introduced.
So far, one 350 hour course has been completed and the educators are engaged in
their courses with teachers. Our experience with using the template has been very
positive in that the educators in training used it to develop scenarios as part of their
course (5 scenarios each) and are now developing more to carry out their own
courses as we can see from the discussions on the portal created by CTI to support
them. So, we share this experience with the ReMath group and present the scenario
template for discussion.
Scenario Template
1. Title
2. Scenario Identity

10. – Writer(s).
– Subject area
– Topic(s) – links to curriculum
3. Activity rationale
– Innovation
– Added value
– Learning problem addressed / theoretical framework
4. Context of implementation
– Who is this addressed to?
– Duration
– Place
– Students’ prerequisites
– Resources and tools
– Class orchestration
– Goals
5. Enactment of the activity
– Enactment of activities
– Tools
– Resources
– Roles and social orchestration
6. Evaluation after the implementation / hints
7. Possible extension
8. References

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